Bayesian Data Analysis

MATH 470 Summary

Sam Turner

Fellingham and Fisher (2017)

Pros: Bayesian model

  • Bayesian model

Fellingham and Fisher (2017) Cont.

Cons:

  • Sparingly uses multilevel modeling

    • Only use multilevel modeling on their orthogonal quartic polynomial

  • Priors are uninformative and don’t fit data

    • Expect average player’s HR probability to be 0.0015

    • Priors effectively suggest that players could have a HR probability between 0 and 1

  • Strange choices of parameters

    • Age, decade of birth, season of play, home ballpark

Model

\[ \begin{align} HR_{nip} &\sim Binomial(AB_{nip},\pi_{nip})\\ \log(\frac{\pi_{nip}}{1-\pi_{nip}}) &= \alpha_{n}+\beta_n\cdot (Age_{ni}-30)+\eta_n\cdot (Age_{ni}-30)^2\\ &+\delta_p+\xi_i \end{align} \]

The amount of HR’s hit by player \(n\) in year \(i\) played at park \(p\) is binomially distributed, according to player \(n\)’s AB’s and HR probability \(\pi\) for year \(i\) at park \(p\).

HR probability \(\pi\) is measured on the logistic scale.

“Innate” HR Hitting Ability - \(\alpha_n\)

\[ \begin{align} \alpha_n&\sim Normal(\mu_0, \sigma_0), n\in\{1,...,657\}\\ \\ \mu_0&\sim Normal(-3.5,0.1)\\ \sigma_0&\sim Exponential(1) \end{align} \]

  • Intercept term which represents “innate” hitting ability of players skilled enough to play in MLB

  • -3.5 on the logit scale is ≈0.029 or 2.9%

  • -3.5±0.1 = [-3.6, -3.4] = [0.0266, 0.0323] = [2.7%, 3.2%]

  • -3.5±2 = [-5.5, -1.5] = [0.0041, 0.182] = [0.4%, 18.2%]

HR Hitting by Age

HR Distribution

HR Distribution Cont.

Centered Age Effect - \(\beta_n\)

\[ \begin{align} \beta_n &\sim Normal(\mu_1,\sigma_1), n\in\{1,...,657\}\\ \\ \mu_1 &\sim Normal(0,0.1)\\ \sigma_1 &\sim Exponential(10) \end{align} \]

  • Multiplicative effect representing how deviation from centered age (Age - 30) affects HR hitting ability

  • Age plays a factor in hitting HR’s, but it is likely not very large so we have the priors set near 0 to reflect this

Centered Age Effect Squared - \(\eta_n\)

\[ \begin{align} \eta_n &\sim Normal(\mu_2, \sigma_2),n\in\{1,...,657\}\\ \\ \mu_2 &\sim Normal(0, 0.01)\\ \sigma_2 &\sim Exponential(100)\\ \end{align} \]

  • Multiplicative effect representing how deviation from centered age squared [(Age - 30)²] affects HR hitting ability

  • Used to capture the non-linearity of the data without risk of over-fitting

HR Hitting by Age

Park Effect - \(\delta_p\)

\[ \begin{align} \delta_p &\sim Normal(\mu_5,\sigma_5),p\in\{1,...,88\}\\ \\ \mu_5 &\sim Normal(0,0.01)\\ \sigma_5 &\sim Exponential(10) \end{align} \]

  • Intercept term which captures the effect playing in different parks has on HR probabilities

  • Parks differ by both dimensions and altitude which affects HR rates

Park Overlay

Park Overlay Cont.

Year Effect - \(\xi_i\)

\[ \begin{align} \xi_i &\sim Normal(\mu_6,\sigma_6),i\in\{1,...,47\}\\ \\ \mu_6 &\sim Normal(0,0.25)\\ \sigma_6 &\sim Exponential(10) \end{align} \]

  • Intercept term which captures the effect playing in different years has on HR probability

  • Changes can occur because of rules, ownership goals, player goals, etc.

  • This term captures those changes without asking why there are changes

HR’s by Year in MLB

HR Proportion by Year in MLB

Model (Again)

\[ \begin{align} HR_{nip} &\sim Binomial(AB_{nip},\pi_{nip})\\ \log(\frac{\pi_{nip}}{1-\pi_{nip}}) &= \alpha_{n}+\beta_n\cdot (Age_{ni}-30)+\eta_n\cdot (Age_{ni}-30)^2\\ &+\delta_p+\xi_i \end{align} \]

  • There are \(1.17\times10^{12}\) parameters for this model

  • The model predicts that the average player’s \(\pi\) is about 0.03 (on the normal scale) which is what we observe in the data

  • Uses Bayesian techniques to update estimates based on what the data says - allows for inference

Robin Yount - Actual \(\pi\)

Robin Yount - Predicted \(\pi\)

Pat Tabler - Actual \(\pi\)

Pat Tabler - Predicted \(\pi\)

Kendrys Morales - Actual \(\pi\)

Kendrys Morales - Predicted \(\pi\)

Goodness of Fit - Trace-plots

Conclusions

  • Will our model make the Hood math department excellent gamblers?

    • No. But it did a fine job at adapting predictions of players and being “right” on average

  • Areas of future research

    • Player archetypes and physical characteristics

    • Considering more for a longer time interval (like Fellingham and Fisher (2017))

    • Better data (advanced metrics or play-by-plays)

Conclusions Cont.

  • We believe we did better than Fellingham and Fisher (2017), but without access to their model and computational resources it is difficult to determine